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mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions. These theories are usually studied ...
, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given range (the ''local'' or ''relative'' extrema), or on the entire
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined ** Domain of definition of a partial function ** Natural domain of a partial function **Domain of holomorphy of a function * ...
(the ''global'' or ''absolute'' extrema). Pierre de Fermat was one of the first mathematicians to propose a general technique, adequality, for finding the maxima and minima of functions. As defined in
set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...
, the maximum and minimum of a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
are the
greatest and least elements In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set (poset) is an element of S that is greater than every other element of S. The term least element is defined dually, that is, it is an el ...
in the set, respectively. Unbounded infinite sets, such as the set of
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s, have no minimum or maximum.


Definition

A real-valued function ''f'' defined on a
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined ** Domain of definition of a partial function ** Natural domain of a partial function **Domain of holomorphy of a function * ...
''X'' has a global (or absolute) maximum point at ''x'', if for all ''x'' in ''X''. Similarly, the function has a global (or absolute) minimum point at ''x'', if for all ''x'' in ''X''. The value of the function at a maximum point is called the of the function, denoted \max(f(x)), and the value of the function at a minimum point is called the of the function. Symbolically, this can be written as follows: :x_0 \in X is a global maximum point of function f:X \to \R, if (\forall x \in X)\, f(x_0) \geq f(x). The definition of global minimum point also proceeds similarly. If the domain ''X'' is a
metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...
, then ''f'' is said to have a local (or relative) maximum point at the point ''x'', if there exists some ''ε'' > 0 such that for all ''x'' in ''X'' within distance ''ε'' of ''x''. Similarly, the function has a local minimum point at ''x'', if ''f''(''x'') ≤ ''f''(''x'') for all ''x'' in ''X'' within distance ''ε'' of ''x''. A similar definition can be used when ''X'' is a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
, since the definition just given can be rephrased in terms of neighbourhoods. Mathematically, the given definition is written as follows: :Let (X, d_X) be a metric space and function f:X \to \R. Then x_0 \in X is a local maximum point of function f if (\exists \varepsilon > 0) such that (\forall x \in X)\, d_X(x, x_0)<\varepsilon \implies f(x_0)\geq f(x). The definition of local minimum point can also proceed similarly. In both the global and local cases, the concept of a can be defined. For example, ''x'' is a if for all ''x'' in ''X'' with , we have , and ''x'' is a if there exists some such that, for all ''x'' in ''X'' within distance ''ε'' of ''x'' with , we have . Note that a point is a strict global maximum point if and only if it is the unique global maximum point, and similarly for minimum points. A
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...
real-valued function with a
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Britis ...
domain always has a maximum point and a minimum point. An important example is a function whose domain is a closed and bounded interval of
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s (see the graph above).


Search

Finding global maxima and minima is the goal of mathematical optimization. If a function is continuous on a closed interval, then by the
extreme value theorem In calculus, the extreme value theorem states that if a real-valued function f is continuous on the closed interval ,b/math>, then f must attain a maximum and a minimum, each at least once. That is, there exist numbers c and d in ,b/math> s ...
, global maxima and minima exist. Furthermore, a global maximum (or minimum) either must be a local maximum (or minimum) in the interior of the domain, or must lie on the boundary of the domain. So a method of finding a global maximum (or minimum) is to look at all the local maxima (or minima) in the interior, and also look at the maxima (or minima) of the points on the boundary, and take the largest (or smallest) one. For
differentiable functions In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point i ...
, Fermat's theorem states that local extrema in the interior of a domain must occur at critical points (or points where the derivative equals zero). However, not all critical points are extrema. One can distinguish whether a critical point is a local maximum or local minimum by using the
first derivative test In calculus, a derivative test uses the derivatives of a function to locate the critical points of a function and determine whether each point is a local maximum, a local minimum, or a saddle point. Derivative tests can also give information ab ...
, second derivative test, or
higher-order derivative test In calculus, a derivative test uses the derivatives of a function to locate the critical points of a function and determine whether each point is a local maximum, a local minimum, or a saddle point. Derivative tests can also give information ab ...
, given sufficient differentiability. For any function that is defined
piecewise In mathematics, a piecewise-defined function (also called a piecewise function, a hybrid function, or definition by cases) is a function defined by multiple sub-functions, where each sub-function applies to a different interval in the domain. P ...
, one finds a maximum (or minimum) by finding the maximum (or minimum) of each piece separately, and then seeing which one is largest (or smallest).


Examples

For a practical example, assume a situation where someone has 200 feet of fencing and is trying to maximize the square footage of a rectangular enclosure, where x is the length, y is the width, and xy is the area: : 2x+2y = 200 : 2y = 200-2x : \frac = \frac : y = 100 - x : xy=x(100-x) The derivative with respect to x is: :\begin \fracxy&=\fracx(100-x) \\ &=\frac \left(100x-x^2 \right) \\ &=100-2x \end Setting this equal to 0 :0=100-2x :2x=100 :x=50 reveals that x=50 is our only critical point. Now retrieve the endpoints by determining the interval to which x is restricted. Since width is positive, then x>0, and since that implies that Plug in critical point as well as endpoints 0 and into and the results are 2500, 0, and 0 respectively. Therefore, the greatest area attainable with a rectangle of 200 feet of fencing is


Functions of more than one variable

For functions of more than one variable, similar conditions apply. For example, in the (enlargeable) figure on the right, the necessary conditions for a ''local'' maximum are similar to those of a function with only one variable. The first partial derivatives as to ''z'' (the variable to be maximized) are zero at the maximum (the glowing dot on top in the figure). The second partial derivatives are negative. These are only necessary, not sufficient, conditions for a local maximum, because of the possibility of a saddle point. For use of these conditions to solve for a maximum, the function ''z'' must also be differentiable throughout. The
second partial derivative test The second (symbol: s) is the unit of time in the International System of Units (SI), historically defined as of a day – this factor derived from the division of the day first into 24 hours, then to 60 minutes and finally to 60 seconds each ...
can help classify the point as a relative maximum or relative minimum. In contrast, there are substantial differences between functions of one variable and functions of more than one variable in the identification of global extrema. For example, if a bounded differentiable function ''f'' defined on a closed interval in the real line has a single critical point, which is a local minimum, then it is also a global minimum (use the intermediate value theorem and Rolle's theorem to prove this by contradiction). In two and more dimensions, this argument fails. This is illustrated by the function :f(x,y)= x^2+y^2(1-x)^3,\qquad x,y \in \R, whose only critical point is at (0,0), which is a local minimum with ''f''(0,0) = 0. However, it cannot be a global one, because ''f''(2,3) = −5.


Maxima or minima of a functional

If the domain of a function for which an extremum is to be found consists itself of functions (i.e. if an extremum is to be found of a
functional Functional may refer to: * Movements in architecture: ** Functionalism (architecture) ** Form follows function * Functional group, combination of atoms within molecules * Medical conditions without currently visible organic basis: ** Functional sy ...
), then the extremum is found using the
calculus of variations The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
.


In relation to sets

Maxima and minima can also be defined for sets. In general, if an ordered set ''S'' has a greatest element ''m'', then ''m'' is a maximal element of the set, also denoted as \max(S). Furthermore, if ''S'' is a subset of an ordered set ''T'' and ''m'' is the greatest element of ''S'' with (respect to order induced by ''T''), then ''m'' is a least upper bound of ''S'' in ''T''. Similar results hold for least element, minimal element and
greatest lower bound In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest ...
. The maximum and minimum function for sets are used in
database In computing, a database is an organized collection of data stored and accessed electronically. Small databases can be stored on a file system, while large databases are hosted on computer clusters or cloud storage. The design of databases ...
s, and can be computed rapidly, since the maximum (or minimum) of a set can be computed from the maxima of a partition; formally, they are self-
decomposable aggregation function In database management, an aggregate function or aggregation function is a function where the values of multiple rows are grouped together to form a single summary value. Common aggregate functions include: * Average (i.e., arithmetic mean) * ...
s. In the case of a general
partial order In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...
, the least element (i.e., one that is smaller than all others) should not be confused with a minimal element (nothing is smaller). Likewise, a greatest element of a
partially ordered set In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...
(poset) is an upper bound of the set which is contained within the set, whereas a maximal element ''m'' of a poset ''A'' is an element of ''A'' such that if ''m'' ≤ ''b'' (for any ''b'' in ''A''), then ''m'' = ''b''. Any least element or greatest element of a poset is unique, but a poset can have several minimal or maximal elements. If a poset has more than one maximal element, then these elements will not be mutually comparable. In a
totally ordered In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexive ...
set, or ''chain'', all elements are mutually comparable, so such a set can have at most one minimal element and at most one maximal element. Then, due to mutual comparability, the minimal element will also be the least element, and the maximal element will also be the greatest element. Thus in a totally ordered set, we can simply use the terms ''minimum'' and ''maximum''. If a chain is finite, then it will always have a maximum and a minimum. If a chain is infinite, then it need not have a maximum or a minimum. For example, the set of
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
s has no maximum, though it has a minimum. If an infinite chain ''S'' is bounded, then the closure ''Cl''(''S'') of the set occasionally has a minimum and a maximum, in which case they are called the greatest lower bound and the least upper bound of the set ''S'', respectively.


See also

*
Arg max In mathematics, the arguments of the maxima (abbreviated arg max or argmax) are the points, or elements, of the domain of some function at which the function values are maximized.For clarity, we refer to the input (''x'') as ''points'' and t ...
*
Derivative test In calculus, a derivative test uses the derivatives of a function to locate the critical points of a function and determine whether each point is a local maximum, a local minimum, or a saddle point. Derivative tests can also give information ab ...
* Infimum and supremum * Limit superior and limit inferior * Mechanical equilibrium * Mex (mathematics) *
Sample maximum and minimum In statistics, the sample maximum and sample minimum, also called the largest observation and smallest observation, are the values of the greatest and least elements of a sample. They are basic summary statistics, used in descriptive statistics ...
* Saddle point


References


External links


Thomas Simpson's work on Maxima and Minima
a
ConvergenceApplication of Maxima and Minima with sub pages of solved problems
* {{Calculus topics Calculus Mathematical analysis Mathematical optimization Superlatives